Theorem elz12s 28863

Description: Membership in the dyadic fractions. (Contributed by Scott Fenton, 7-Aug-2025.)

Assertion

Ref	Expression
elz12s	⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem elz12s

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3471	. 2 ⊢ (𝐴 ∈ ℤs[1/2] → 𝐴 ∈ V)
2		id 23	. . . . 5 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 = (𝑥 /su (2s↑s𝑦)))
3		ovex 7453	. . . . 5 ⊢ (𝑥 /su (2s↑s𝑦)) ∈ V
4	2, 3	eqeltrdi 2868	. . . 4 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V)
5	4	a1i 11	. . 3 ⊢ ((𝑥 ∈ ℤs ∧ 𝑦 ∈ ℕ0s) → (𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V))
6	5	rexlimivv 3204	. 2 ⊢ (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)) → 𝐴 ∈ V)
7		eqeq1 2764	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ 𝐴 = (𝑥 /su (2s↑s𝑦))))
8	7	2rexbidv 3227	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦)) ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦))))
9		df-z12s 28806	. . 3 ⊢ ℤs[1/2] = {𝑧 ∣ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝑧 = (𝑥 /su (2s↑s𝑦))}
10	8, 9	elab2g 3633	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦))))
11	1, 6, 10	pm5.21nii 381	1 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450  (class class class)co 7420   /_su cdivs 28578  ℕ_0scn0s 28703  ℤ_sczs 28769  2_sc2s 28801  ↑_scexps 28803  ℤ_s[1/2]cz12s 28805

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rex 3087  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4585  df-pr 4587  df-uni 4867  df-iota 6494  df-fv 6546  df-ov 7423  df-z12s 28806

This theorem is used by:  elz12si  28864  zz12s  28866  z12no  28867  z12addscl  28868  z12negscl  28869  z12shalf  28871  z12zsodd  28873  z12sge0  28874